Assume that the mouse has equal probabilities to enter room 1, 2 or 3.
Let X be the expected time (in minutes).
If mouse enters room 1 (prob = 1/3): 3 minutes;
If mouse enters room 2 (prob = 1/3): 4 + X minutes;
If mouse enters room 3 (prob = 1/3): 5 + X minutes;
Solving X = 1/3 * 3 + 1/3 * (4 +X) + 1/3 * (5 + X), we get X = 12.